Analysis of a defective renewal equation arising in ruin theory. (English) Zbl 1028.91556

Summary: This paper studies in detail the solution of a defective renewal equation which involves the time of ruin, the surplus immediately before ruin, and the deficit at the time of ruin. The analysis is simplified by introduction and analysis of a related compound geometric distribution, which is studied in detail. Tijms approximations and bounds for these quantities are also discussed. Examples are given for the cases when the claim size distribution is exponential, combinations of exponentials and mixtures of Erlangs. In a subsequent paper, we will extend our analysis to the moments of the time of ruin, the moments of the surplus before the time of ruin, the moments of the deficit at the time of ruin, and correlations between them.


91B30 Risk theory, insurance (MSC2010)
60K05 Renewal theory
Full Text: DOI


[1] Bowers, N., Gerber, H., Hickman, J., Jones, D., Nesbitt, C., 1997. Actuarial Mathematics, Second Edition. Society of Actuaries, Itasca, IL.
[2] De Vylder, F.E., 1996. Advanced Risk Theory: A Self-Contained Introduction. Editions de L’Universite de Bruxelles, Brussels.
[3] De Vylder, F.E.; Goovaerts, M.J., Discussion of ‘the time value of ruin’ by gerber and shiu, North American actuarial journal, 2, 72-74, (1998)
[4] Di Lorenzo, E.; Tessitore, G., Approximate solutions of severity of ruin, Blatter deutsche gesellschaft fur versicherungsmathematik, XXII, 705-709, (1996) · Zbl 0860.62078
[5] Dickson, D.C.M.; Dos Reis, A.D.S.; Waters, H.R., Some stable algorithms in ruin theory and their applications, ASTIN bulletin, 25, 153-175, (1995)
[6] Dickson, D., On the distribution of surplus prior to ruin, Insurance: mathematics and economics, 11, 191-207, (1992) · Zbl 0770.62090
[7] Dickson, D., Dos Reis, A.D.E., 1996. On the distribution of the duration of negative surplus. Scandinavian Actuarial Journal, 148-164. · Zbl 0864.62069
[8] Dickson, D., Waters, H.R. 1992. The probability and severity of ruin in finite and in infinite time. ASTIN Bulletin 22, 177-190.
[9] Delbaen, F., A remark on the moments of ruin time in classical risk theory, Insurance: mathematics and economics, 9, 121-126, (1990) · Zbl 0733.62108
[10] Dufresne, F.; Gerber, H., The probability and severity of ruin for combinations of exponential claim amount distributions and their translations, Insurance: mathematics and economics, 7, 75-80, (1988) · Zbl 0637.62101
[11] Fagiuoli, E.; Pellerey, F., Preservation of certain classes of life distributions under Poisson shock models, Journal of applied probability, 31, 458-465, (1994) · Zbl 0806.60075
[12] Feller, W., 1971. An Introduction to Probability Theory and Its Applications, vol. 2, 2nd ed. John Wiley, New York. · Zbl 0219.60003
[13] Gerber, H., 1979. An Introduction to Mathematical Risk Theory. S.S. Huebner Foundation, University of Pennsylvania, Philadelphia. · Zbl 0431.62066
[14] Gerber, H.; Goovaerts, M.; Kaas, R., On the probability and severity of ruin, ASTIN bulletin, 17, 151-163, (1987)
[15] Gerber, H.; Shiu, E.S.W., The joint distribution of the time of ruin, the surplus immediately before ruin and the deficit at ruin, Insurance: mathematics and economics, 21, 129-137, (1997) · Zbl 0894.90047
[16] Gerber, H., Shiu, E.S.W., 1998. On the time value of ruin. North American Actuarial Journal 2, 48-72 and 72-78. · Zbl 1081.60550
[17] Gerber, H.; Shiu, E.S.W., From ruin theory to pricing reset guarantees and perpetual put options, Insurance: mathematics and economics, 24, 3-14, (1999) · Zbl 0939.91065
[18] Goulden, I.P., Jackson, D.M., 1983. Combinatorial Enumeration. Wiley, New York.
[19] Hesselager, O., Wang, D., Willmot, G.E., 1998. Exponential and scale mixtures and equilibrium distributions. Scandinavian Actuarial Journal, 125-142. · Zbl 1076.62559
[20] Lin, X., Tail of compound distributions and excess time, Journal of applied probability, 33, 184-195, (1996) · Zbl 0848.60081
[21] Lin, X., Willmot, G.E., 1999. The moments of the time of ruin, the surplus before ruin and the deficit at ruin, in preparation. · Zbl 0971.91031
[22] Massey, W.; Whitt, W., A probabilistic generalisation of taylor’s theorem, Statistics and probability letters, 16, 51-54, (1993) · Zbl 0765.60032
[23] Neuts, M., 1981. Matrix-Geometric Solutions in Stochastic Modeling: An Algorithmic Approach. The John Hopkins University Press, Baltimore. · Zbl 0469.60002
[24] Panjer, H.H., Willmot, G.E., 1992. Insurance Risk Models. Society of Actuaries, Schaumburg, IL.
[25] Picard, Ph.; Lefevre, C., The moments of ruin time in the classical risk model with discrete claim size distribution, Insurance: mathematics and economics, 23, 157-172, (1998) · Zbl 0957.62089
[26] Resnick, S., 1992. Adventures in Stochastic Processes. Birkhauser, Boston. · Zbl 0762.60002
[27] Shiu, E.S.W., Calculation of the probability of eventual ruin by beekman’s convolution series, Insurance: mathematics and economics, 7, 41-47, (1988) · Zbl 0664.62112
[28] Steutel, 1970. Preservation of Infinite Divisibility under Mixing and Related Topics. Math. Centre Tracts 33, Math. Centre, Amsterdam. · Zbl 0226.60013
[29] Tijms, H., 1986. Stochastic Modelling and Analysis: A Computational Approach. Wiley, Chichester.
[30] Tijms, H., 1994. Stochastic Models: An Algorithmic Approach. Wiley, Chichester. · Zbl 0838.60075
[31] Willmot, G., Further use of shiu’s approach to the evaluation of ultimate ruin probabilities, Insurance: mathematics and economics, 7, 275-282, (1988) · Zbl 0675.62075
[32] Willmot, G., Refinements and distributional generalizations of lundberg’s inequality, Insurance: mathematics and economics, 15, 49-63, (1994) · Zbl 0814.62070
[33] Willmot, G.E., On a class of approximations for ruin and waiting time probabilities, Operations research letters, 22, 27-32, (1997) · Zbl 0911.90172
[34] Willmot, G.E., Bounds for compound distributions based on Mean residual lifetimes and equilibrium distributions, Insurance: mathematics and economics, 21, 25-42, (1997) · Zbl 0924.62110
[35] Willmot, G.E.; Lin, X., Simplified bounds on the tails of compound distributions, Journal of applied probability, 34, 127-133, (1997) · Zbl 0882.60088
[36] Willmot, G.E.; Lin, X., Exact and approximate properties of the distribution of surplus before and after ruin, Insurance: mathematics and economics, 23, 91-110, (1998) · Zbl 0914.90074
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.